Alternative solutions of the black-sholes equation
In the mathematical treatment of financial derivatives, and specially that of options, the defining stochastic differential equation coupled with the arbitrage-free pricing condition leads to a deterministic partial differential equation. The solution of this equation under appropriate boundary conditions is interpreted as the price of the asset. A review of the specialized financial literature reveals that, by and large, there are three main altrernative approaches to finding the function that describes the price of the contingent claim. Fist, we have the classical methods of solution of partial differential equations. These methods are widely used in phusics, but they have yet to take root in the economic arena. Their importance, however, cannot be overlooked, and we note in passing that even in the simplest cases the solution of a partial differential equation is everthing but trivial. A second approach can be found in the mehods of mathematical statistics, where the stochastic differential equation describing the asset price is solved via the equivalence between the financial no-arbitrage condition and that of a martingale. Both of these approaches often lead to closed-form solutions that are easy to use and convenient for the valuation of assets in real time. Finally, when a solution in closed form is not available or possible, numerical methods may provide an alternative solution. Nevertheless, the application of these methods in finance still presents great practical difficulties, due mainly to the time involved in obtaining a solution. This paper deals with the first of the approaches mentioned above. Using the Black-Scholes equation for the valuation of european options, a fairly detailed and simple description is given of some of the methods that can be used to solve this type of equation.
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